1 What is technology?

A few hundred years ago, pencil and paper were technology. Some time later, the slide rule was technology. Chalk and blackboards were technology. Today, computer algebra systems and the Internet are technology.

How has the advance in technology affected the way we teach and learning of mathematics? In many ways, we seem to have reached an impasse. Pencil and paper are still the tools students use to transcribe chalk-and-blackboard lectures into notebooks.

Perhaps the reference to chalk-and-blackboard delivery is unfair. As Figure 1 shows, whiteboards and marker pens have often replaced blackboards and chalk. Progress? Not really.

Some professors have access to "smartboards." That's progress. A smartboard is a large computer touch screen driven by a computer program that lets professors write, draw and save their lectures electronically. The results are computer files that can often be freely circulated and reworked. How the material can be manipulated depends on the software involved. Examples of the effective use of mathematical software in the teaching of linear algebra may be found in Szabo [9,11], and methodological issues involved were explored in Szabo [12,13], and in numerous national and international workshops. Complementary traditional study methods are discussed in Szabo [7].

Figure 1: Students taking notes during a traditional lecture

1.1 Computer algebra systems

Among the best-known computer programs for doing mathematics are Mathematica, Maple, and MuPAD. Whether and to what extend a university uses any or all of these systems for teaching and learning depends on the affluence of the university, the personal preferences of its professors, and the actual use of the systems in the classroom.

At some level, all of these systems provide the same functionality. You can use them as sophisticated calculators and as devices for minimizing the arithmetic required to solve problems in calculus, linear algebra, and other subjects. But you can also use them as tools for visualizing mathematical relationships and properties of functions, sets, and other mathematical objects (see Majewski [5,6]). However, none of these systems provide the look-and-feel of live mathematics.

Example 1: Using Mathematica to calculate a standard deviation

Here are the steps used to calculate the standard deviation of the list of numbers [4,2,9].

Here are the equivalent steps in Maple.

Example 2: Using Maple to calculate a standard deviation

In this example, we can see that in Maple, the function required to calculate the standard deviation of the list of numbers [4,2,9] requires the stats package, and that the function has a two options to distinguish between the "population" and "sample"versions of the standard deviation.

The MuPAD calculations are similar to those in Maple and we refer to programming guides Majewski ([5,6]) for details. Additional relevant comments may be found in Szabo ([8,10]) .

In the two given examples, we have illustrated what can be achieved by "pressing buttons." The user defines the data and inserts them in appropriate functions, specified in menus, and clicks the Execute button. No knowledge or understanding is required to produce a numerically correct result.

One of the criticisms advanced against the use of computer algebra systems in teaching and learning is that the only thing students learn is how to select functions from menus to produce results they do not understand and cannot explain. Why? The mathematics is missing from the process. This is one of the main obstacles to the successful use of computer algebra systems in the classroom.

1.2 The Scientific Notebook interface to computer algebra

There is really only one truly elegant and natural electronic whiteboard for the teaching and learning of mathematics. It is called Scientific Notebook. It is the affordable cousin of Scientific WorkPlace, a system that combines the formatting language LaTeX and the computer algebra system MuPAD into one elegant visually rich environment. The teaching and learning of mathematics with Scientific Notebook at the undergraduate level and, to some extent, at the postgraduate level as well, is easy and has considerable potential for advancing the cause of mathematics.

Let us illustrate this remark by showing how the teaching and learning of a mathematical concept can be facilitated computationally and visually using Scientific Notebook.

Suppose we that we are trying to introduce the concept of a standard deviation. As in Mathematica and Maple, we can use the built-in function to compute the standard deviation of a list of data.

Example 3: Anatomy of a formula

As in the case of Mathematica and Maple, we can calculate the standard deviation of the list of numbers [4,2,9] by pressing a button. Pointing the cursor to the given list and selecting the function

Compute > Statistics > Standard Deviation

produces the result

[4,2,9], Standard deviation(s):

However, we can produce the same result by building up the standard deviation formula step-by-step and practice the use of dot products, sums, averages, normalization, and square roots, and variance at the same time.

1. 

2. 

3. 

4. 

In step 1, we find the average of the given values. In step 2, we normalize the given values around the origin. In step 3 we find the variance of the normalized values, elegantly expressed using the dot product of column vectors, and in step 4, we take the square root of the variance.

We can now copy the left-hand side of the equation in steps 3 and change 1/2 to 1/3, and calculate the population" version of the standard deviation:

1. 

2. 

We have used Scientific Notebook to develop a mathematically rich structure for studying the formulas for standard deviations.

Although we could have done all of this using pencil and paper, or chalk and blackboard, the Scientific Notebook approach has several advantages:

• No information is lost.

• We can revisit all of the steps at any time and as often as we like.

• We can recycle each line of our calculations.

• All arithmetic steps involved are correct and no time is lost with frustrating corrections.

• We can modify the steps involved and explore the results.

• We can save time performing routine calculations and concentrate on ideas and concepts instead.

1.3. From convenience to necessity

There are topics in undergraduate mathematics that cannot easily be taught and learned without technology. A good example is that of the eigenvalues of real matrices. While it is straightforward to motivate and define eigenvalues as the roots of characteristic polynomials, the absence of an algorithm for finding the roots of real polynomials of degree five or higher makes it almost impossible to give a meaningful presentation of the subject. Computer algebra systems, on the other hand, have built-in tools for approximating eigenvalues and their graphic capabilities make it easy to plot the graphs of characteristic polynomials.

Example 4: The eigenvalues of a real symmetric 5x5 matrix

Let be

a real 3x5 matrix. Then the matrix

 

is symmetric and we know from linear algebra that its characteristic polynomial has real roots.

Let us use Scientific Notebook to try to find them.

,

characteristic polynomial:

and

, roots:

As we can see, the result produces five approximations of the real roots of the characteristic polynomial of the matrix A^TA The result also shows that if we plot p(X), we need to plot it on an interval including 0 and 450 if we expect to provide a graphic illustration of the fact that the polynomial has five real roots. Since 0 is a repeated root, we expect the graph of p(X) to cross the X -axis only four times. However, unless we have a large screen, any attempt to visualize the roots of p(X) on the graph of the polynomial is difficult. We therefore break the required interval into two pieces: points from -10 to 30, and points from 30 to 460 for example, and plot p(X) on the two adjoining intervals:

 

Example 5: The singular value decomposition of a rectangular matrix

One of the most useful linear algebra tools for manipulating graphic images, considered as matrices, is the singular value decomposition. It breaks a real mxn matrix A into a product UDV^T with U and V orthogonal and D either diagonal or a diagonal matrix augmented by zero rows and columns. Working the matrices U,D and V is usually considerably simpler than working with the matrix A. For example, an application may require the inverse of A. It may be simpler to compute the inverse of UDV^T than the inverse of A since we can reduce U,D, and V^T to U_rD_rV_r^T using the rank r of A so that D_r is actually diagonal. Using these components, we can now define the matrix

If A is invertible, then A⁺=A^-1. Otherwise A⁺, known as the pseudo-inverse of A can be used in many cases as a close substitute for A^-1.

Singular value decomposition is relatively easy to discuss using the visual features of Scientific Notebook. It can also be discussed using Mathematica, Maple, and MuPAD directly, but beginning students often find the command-line presentation less convincing.

1.4 Lectures and computer algebra

Replacing blackboards and chalk and whiteboards and marker pens with smartboards and computer algebra systems certainly facilitates and enriches teaching and learning for the reasons illustrated. However, in today's world, even the most elegant electronically enriched lectures suffer from a variety of shortcomings.

In a recent interview for Maclean's Magazine [4], Alison Gopnik, a cognitive scientist at Berkeley University puts it bluntly: Just stop lecturing them. She maintains that the way people learn best is by effectively interacting with their environment. She says that there is a staggering contrast between what I know about learning from the lab, and the way I teach in the classroom. She feels that the best learning takes place through guided apprenticeship, when a learner tries different things, with an expert providing feedback, much in the way sports and music are taught.

Valérie Bouchard, a twenty-year-old student in Montréal, agrees. In [3] to Gopnik's interview in [3], she writes that it is impossible for a student to memorize all the information when the teacher speaks around two or three hours in a row. While the teacher is lecturing, the students are passive, so they tend to fall asleep or stop listening. When I'm in class, I can concentrate for forty-five minutes to one hour and after that, I take notes, but I do not remember the theory after the class, so I have to study harder before the exam. So, I learn better when the teacher takes a break and make us do exercises to practice what we have just learned.

It is of course true that teachers have not been standing still. Most teachers have introduced some elements of technology into their courses. In their highly practical and useable how-to book entitled Teaching Engineering, Wankat and F. S. Oreovicz (see [15]) of the University of Purdue have analyzed what is wrong with traditional lectures and have indicated steps for remedying some of the concerns voice by students. One suggestion is to use lab-based technology to enrich the teaching and learning process.

Mike Wong of the University of Washington, in his article entitled My students looked bored in class (see Wong [16]), makes several suggestions for making lectures more interactive and personable, in the context of the university's Narratives Supporting Excellence in Teaching program. Some of these suggestions could benefit from technology.

Wong suggest that we should make it easier for our students to stay focused by breaking up the lecture with activities. Consider assigning in-class problem solving to pairs, small groups, or the whole class, and then walk around the classroom to help. Also students could provide feedback to their peers, field questions, defend their answers, provide real world example, or write test questions.

This is precisely what takes place in my lab-based courses and is discussed in more detail in my comments below on the nature and relevance of the tutorial method as a tool for general teaching and learning.

A more radical departure from the lecture model has been implemented by some mathematicians. In [14], Professor Uhl of the University of Illinois at Urbana-Champaign explains how technology has led him to stop lecturing and start teaching.

He writes that I can see that over the years my lecturing style and techniques evolved to be remarkably similar to those of today's good lecturers---enthusiasm, humor, content. I was a very popular lecturer and recently won an MAA sectional award for distinguished teaching based in no small part on the lecture courses I gave at Illinois between 1968 and 1988. But for the last ten years, I have completely abandoned the long lecture method.

He also points out that with technology at the beginning of the learning process, we can get students intrigued by seeing the math comes to life before the strange language and the proofs go on. ...Good use of technology is a great motivator. Get the students' interest and then get out of their way---constantly supporting (but never pulling) them as they advance.

2. From lectures to teaching labs

When I was an undergraduate at Oxford University, lectures played a marginal role in our education. Or maybe I should put it the other way around: they played a pivotal role, but not in the sense in which we have been speaking of them. Lectures were given by leading researchers reporting on their latest findings and that of their prominent collaborators at other universities. Attendance was optional.

The central forum for teaching and learning was the tutorial. On my first day of class, I was given a list of tutors whom I had to meet once a week, either alone or with two or three fellow undergraduates, one tutor per course. In each course, our first tutorial was a kind of orientation session. We were told what the course was all about and were assigned a set of problems to be solved by the following week, together with a reading list and instructions on what was expected of us for the rest of the term.

Each tutorial consisted of several elements:

A set of problems to be solved, accompanied by a reading list involving textbooks and journal articles; a motivational introduction to the topic of the week; suggestions of activities for acquiring and knowledge and skills needed to solve the assigned problems; and in subsequent tutorials, an analysis of our solutions, suggestions for corrections and improvements, and the assignment of a new set of problems. Our tutors were our mentors and we, the undergraduates, were the apprentices.

Given this background and my conviction that students learn best by doing, it was natural for me, when technology was sufficiently advanced about fifteen years ago, to embrace it with enthusiasm and incorporate it fully into my "lectures" at the time.

Like Professor Uhl, I have not touched a piece of chalk in the last fifteen years and currently teach all of my courses in the Arts and Science Learning Center teaching labs.

Figure 4: Concordia University Teaching Labs

 

I use Scientific Notebook as my electronic blackboard and chalk, complemented with Maple for detailed computations, programming, and structural analysis.

From the very beginning, the reaction of my students to my style of teaching has been enthusiastic and favorable. "All courses should be taught like this!'' is a frequent comment on my course evaluations.

Here is a snapshot of one of the interactive components of my teaching activities that bridges the gap between the old and new ways of teaching and learning:

Figure 5: A dot product drill with Scientific Notebook

 

It shows how the Note feature of Scientific Notebook can be use to practice the learning of fundamental skills and concepts. In this case, the drill deals with the calculation of the dot product of vectors. The students really enjoy these activities. They cover both theoretical and computational topics. It is therefore not surprising that the failure rate in my courses is low and the enthusiasm and commitment to learning are high. My experience and that of others, as discussed by Uhl, Wong, and Wankat in [14], [16], [15], for example, shows that learning with the help of technology can be made to be intellectually rewarding and pedagogically sound, as long as we keep focused on the fact that technology is a means to an end, not the end itself.

3. From teaching labs to laptops

The big question remains how we can we turn things around and cajole professors, young and not so young, into adopting technology as their medium?

The first step is often the supply of laptops to faculty member at a price. The price is the requirement to use Internet-based course management systems such as WebCT, Moodle, and similar systems, to organize courses and interact with the university and students using these systems.

The use of an electronic course management system always provides opportunities for redesigning course material and reworking lecture note according to new principles. In the case of mathematics, it also provides an ideal opportunity for the creative use of computer algebra.

Figure 6: Screen shot of a web page introducing the online linear algebra course

At McGill University in Montreal, the Department of Education spearheaded a broader approach. By linking educational development and course design to overall university goals and priorities, researchers in educational technology were able to convince the university that it should support the formation of "learning teams," groups of professors in possibly different disciplines, whose mission is to promote the use technology in classrooms throughout the university. Janette Barrington (see [1]) has just completed her PhD, based on this research and has demonstrated the feasibility of this approach to the increase of the use of technology in teaching and learning at the university.

To practice what I preach, I am currently completing the development of an online introductory linear algebra course, using Scientific Notebook as the technological core. The course will be offered by Concordia University starting in January 2007, and will consist of twenty electronic lessons, activities, and feedback mechanisms.

Let me conclude by saying that at Concordia University in Montreal, I am actively involved in the Teaching and Learning Services Center, where professors are exploring different approaches to the improvement of teaching and learning at the university. We have always been known for having small classes, for example. Some of us are hoping to continue this tradition by increasing the use of technology in tutorial settings.

In contrast to the teaching teams approach used by McGill University to promote innovative teaching and learning, the Concordia center uses community-based workshops to promote good and creative teaching, concentrating on the reorganization of existing courses and the design of new courses using conceptual visualization and technological innovation (see Barrington [2]). This requires a level of interdisciplinary collaboration and cooperation that bodes well for the harmonious coexistence of lectures, labs and laptops in the future of the university.

References

1. J. Barrington, Learning Teams: A communities-of-practice approach to faculty development and university course design, PhD Thesis, Department of Education, Concordia University, Montreal, August 2006.
2. J. Barrington, B. Lazar, and F. E. Szabo, Classrooms, Concepts, and Connections, ICED 2006, Sheffield Hallam University, UK, June 11, 2006.
3. V. Bouchard, Just quit lecturing them, http://www.creativeval.blogspot.com, May 2006.
4. K. MacKlem, Just quit lecturing them, Maclean's, Toronto, Canada, March 27th, 2006.
5. M. Majewski and F. E. Szabo, Developing Open and Flexible Computing Environments for Teaching Mathematics and Science, Proceedings of the 2nd International Conference on the Teaching of Mathematics, Crete, Greece, July 2002.
6. M. Majewski and F. E. Szabo, Integrating MuPAD in the Teaching of Mathematics, Fifth International Conference on Technology in Mathematics Teaching, Klagenfurt, Austria, August 2001.
7. F. E. Szabo, Actuaries' Survival Guide, Elsevier Academic Press, Boston, 2004.
8. F. E. Szabo, Preface, in: M. Majewski, MuPAD Pro Computing Essentials, 2nd Edition, Springer Verlag, Berlin, 2004.
9. F. E. Szabo, Linear Algebra: An Introduction Using Maple, Harcourt Academic Press, Boston, 2002.
10. F. E. Szabo, Preface, in: M. Majewski, MuPAD Pro Computing Essentials, Springer Verlag, Berlin, 2002.
11. F. E. Szabo, Linear Algebra: An Introduction Using Mathematica, Harcourt Academic Press, Boston, 2000.
12. F. E. Szabo, Online Teaching and Learning of Mathematics: Reflections on a New Pedagogy, International Seminar on Distance Education, Macau, China, 2000.
13. F. E. Szabo, Making Mathematics Accessible---Scientific WorkPlace, the Syntax-free Electronic Blackboard, Primer Encuentro Internacional Sobre la Ensenanza y Aplicacion de las Matematicas, ITESM, Mexico, April 1996. (Reprinted at: http://www.mackichan.com)
14. J. Uhl, How technology influenced me to stop lecturing and start teaching, Technology and NCTM Standards 2000,National Council of Teachers of Mathematics, Virginia, USA, June 1998.
15. P. C. Wankat and F. S. Oreovicz, Teaching Engineering, McGraw Hill College, New York, 1998.
16. M. Wong, My students looked bored in class, NEXT: Narratives Supporting Excellence in Teaching, University of Washington, August 2006.

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